The derived superalgebra of skew elements of a semiprime superalgebra with superinvolution

TL;DR

This paper explores the Lie algebra structure of skew elements in a semiprime superalgebra with superinvolution, revealing conditions under which Lie ideals relate to the algebra's decomposition or centrality.

Contribution

It characterizes the Lie ideals of the derived superalgebra of skew elements, linking their structure to the algebra's decomposition or centrality properties.

Findings

If U is a Lie ideal, then either it contains a nonzero ideal of a certain form or the algebra decomposes with U's image being central.

The algebra can be expressed as a subdirect sum involving simple superalgebras of limited dimension.

The structure of Lie ideals is closely tied to the algebra's decomposition and central elements.

Abstract

In this paper we investigate the Lie structure of the derived Lie superalgebra [K, K], with K the set of skew elements of a semiprime associative superalgebra A with superinvolution. We show that if U is a Lie ideal of [K, K], then either there exists an ideal J of A such that the Lie ideal [J \cap K,K] is nonzero and contained in U, or A is a subdirect sum of A', A'', where the image of U in A' is central, and A'' is a subdirect product of orders in simple superalgebras, each at most 16-dimensional over its center.